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The Spin Bivector and Zeropoint Energy in Geometric Algebra

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The Spin Bivector and Zeropoint Energy in Geometric Algebra Adv. Studies Theor. Phys., Vol. 6, 2012, no. 14, 675 - 686 The Spin Bivector and Zeropoint Energy in Geometric Algebra K. Muralidhar Physics Department, National Defence Academy Khadakwasla, Pune-411023, Maharastra, India [email protected] Abstract In a classical argument, treating action variable of an oscillator as a transformed angular momentum, it has been shown that the fluctuations induced on an electron by zeropoint fields produce rotations defined by an imaginary zeropoint angular momentum. The electron spin bivector is identified with this zeropoint angular momentum. The magnitude of spin is estimated from the relation between average zeropoint angular momentum and average zeropoint energy. The bivector nature of electron spin angular momentum is shown purely on classical grounds using geometric algebra. The spin equation and spin precession are presented in the light of geometric algebra. Keywords : Classical spin, Zeropoint energy, Geometric algebra 1. Introduction Historically, the spin angular momentum was hypothesized by Uhlenbeck and Goudsmit [1] and independently by Bichowsky and Urey [2]. However, the existence of spin can be derived from the fundamental postulates of quantum mechanics and the property of symmetry transformations. The spin angular momentum is a kinematic property of massive elementary particles and it corresponds to rotation group symmetry SU(2). The Dirac theory of electron automatically includes spin as a quantum relativistic effect. The Dirac electron executes very rapid oscillations in addition to the uniform rectilinear motion and this oscillatory motion is called zitterbewegung [3]. Thus within the wave packet associated with the electron there exists a superposition of violent oscillations 2 each with angular frequency equal to 2m ec / ħ, where, me is the mass of electron, c is the velocity of light and ħ is the Plank’s constant. This frequency may be 676 K. Muralidhar regarded as the frequency of rotation of the zitterbewegung within the wave packet. Many authors over the years proposed several phenomenological models to explain the spin angular momentum of electron and its dynamical behavior. Huang [4] investigated the Dirac electron and analyzed the spin as an angular momentum of zittrerbewegung circulatory motion. Barducci, Casalbuon and Lusanna [5] investigated path integrals for fermions in Grassmann algebra and considered those paths as spin. Barut and Brachen [6] proposed the spin as the orbital angular momentum associated with the internal oscillatory system. In the approach of geometric algebra, using a multivector valued Lagrangian, Barut and Zanghi [7] arrived at the bivector form of classical internal spin of Dirac electron and opined the spin as the angular momentum of zitterbewegung and in the extensions of semiclassical theories of Dirac electron the spin was identified with a bivector and the point particles execute circular motion by absorbing energy from vacuum field [8, 9]. In the multivector formalism of Pauli theory Hestenes and Gurtler [10] showed that the spin angular momentum is fundamentally a bivector quantity and concluded that in the absence of magnetic fields, the Schrödinger theory is identical with the Pauli theory and the constant imaginary factor i ħ is exactly twice the spin angular momentum. Further in a similar manner, in the hidden geometric structure of Dirac theory, Hestenes [11-13] observed the spin as the imaginary factor iħ in the Dirac equation. The spin has been identified as a local circulatory motion of electron so that spin is connected to the zitterbewegung rotation within the wave packet. More recently by considering the physical space generated from a pair of annihilation and creation operators, the classical origin of spin half fermions was suggested by Baylis, Cabrera and Keselica [14]. The spin was recognised as a physical (but intrinsic) rotation with rotation rate equal to the zitterbewegung frequency. The magnitude of spin has been determined from the g-factor. In the stochastic electrodynamics approach to zeropoint field, we have an explanation to the origin of quantum behavior of matter [15, 16]. As such, it is expected that the nature of spin may be explored in the same treatment. In the theory of classical stochastic electrodynamics of microscopic phenomena Moor and Ramirez [17] discussed phenomenological aspects of spin. Considering angular momentum fluctuations, the electron spin was studied by de la Pena and Jauregui [18] and in this model the spin has been assumed to be hidden in the averaging process and hence the set of realizations of the field are divided into two mutually exclusive and complementary sub ensembles. Then the average of each sub ensemble may contain the components of spin angular momentum. Considering two independent circular polarization states of the electric field vector of zeropoint field, they arrived at an expression for angular momentum and obtained the spin value ħ/2 multiplied by a factor 3/4. The core of semiclassical theories of electron spin, though the spin was derived as a bivector quantity, contains the quantum behaviour in the form of transforming quantum mechanical equations into spacetime algebraic formalism and the magnitude of spin was either introduced or derived from the known quantities from quantum mechanics. However, all the above mentioned theoretical models of classical spin firmly suggest that the origin of spin arises mainly due to the rotation of the electron produced by the absorption of energy from the Spin bivector and zeropoint energy 677 fluctuating zeropoint field. In the case of stochastic theory the magnitude of spin was derived with an additional multiplicative factor. In the previous article on classical origin of quantum spin [19] we have shown that a charged particle immersed in the ZPF may be considered to possess complex oscillations and on purely mathematical grounds such complex oscillations are described in terms of modes of rotation r +(t) and r − (t). The imaginary part of complex rotations is considered in the classical explanation of quantum spin. The basic reason of considering such complex rotations is explained in the present paper. The main aim of the preset paper is to show the bivector nature of spin angular momentum of electron, its geometrical meaning as spin bivector without any quantum background and to derive the correct magnitude of spin. This is achieved by considering the action variables of a periodic system and considering the zero point energy of electron as perturbation to a classical Hamiltonian of an oscillator, it has been shown that the energy absorbed by an electron from zero point fields actually produces rotations rather than oscillations and such rotations correspond to imaginary zeropoint angular momentum which is derived in section 2. As the zero point fields are random fields, the average action variable at zero temperature in relation to the average zero-point energy is presented in section 3. Certain essential aspects of geometric algebra are presented in section 4. The spin is identified with this angular momentum and the geometric meaning of its imaginary nature in terms of geometric algebra is used to express spin bivector. The bivector nature of spin is explored in section 5 and conclusions are presented in section 6. 2. Zero point angular momentum and rotation modes of electron In the periodic motion of a system, a set of integration constants α that appear in the Hamilton Jacobi equation can be suitably defined as an action variable [20]. For a conservative system the momentum can be expressed in terms of integration constants α, as p = p(x, α) and this equation traces out a curve in two dimensional phase space. The periodic motion of a system represents oscillations (librations), if the system point traces out a closed orbit in the phase space. The values of position coordinate x are bounded and the initial position lies in two zeros of kinetic energy. However, when the system point traces out a periodic curve or an open orbit in two dimensional phase space the system represents rotations such that a 2 π rotation keeps the system unchanged. The values of x are unbounded and increase indefinitely. In either case one can introduce an action variable I as a transformed constant angular momentum [21]. 1 I = ∫ p dx (1) 2π Where, the integration is to be carried out over a complete period of oscillation or rotation. An electron motion in fluctuating random zeropoint field can be approximated as simple harmonic. In addition to zero point fields, there exists thermal radiation and the energy of the electron oscillator can be considered as 678 K. Muralidhar sum of two parts, one as thermal energy E k and the other the zero point energy ∆E0 associated with the electron. Then Hamiltonian of the electron oscillator is H 2 2 = E k + ∆E0 + V(x), where potential energy V(x) = ½ m e ω0 x . Since ∆E0 arises from the fluctuating fields, the Hamiltonian describes the motion of a nonlinear harmonic oscillator. First let us consider the Hamiltonian in the absence of zero 1/2 point fields, H0 = E = E k + V(x) and the momentum p = ± [2m e (E – V)] . The phase space diagram of the system gives a closed elliptical orbit under the condition E – V < 0, the harmonic oscillator is said to be in oscillation or libration mode and the area enclosed by the curve divided by 2 π is equal to the action variable I. When the kinetic energy is too large i.e., under the condition E – V > 0, there is a possibility that the system point traces out a periodic curve or open orbit and the system is said to be in rotation mode. An important situation arises when the maximum energy of the oscillator equals the potential energy. The system point represents two saddle points in the phase space corresponding to two zeros of kinetic energy and the curve passing through these points is called separatrix.
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